M3L12c.txt

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There are situations where the leading approximation vanishes.
For instance, if two levels have the same parody,
then the dipole operator between the two of them is 0.
And then if you want to have a transition between--
or if you want to consider transition between those two
levels, it will come from next order terms.
So let's now discuss higher order radiation processes.

So the motivation of why I want to discuss higher order
radiation processes is because in some basic courses,
you only need the dipole approximation
and you think the dipole coupling
is the only coupling which exists in the world.
And by going to high order, I want to show you
this is not the case.
Also, I wanted to give you an idea what
it means if you have a leading order,
transitions are forbidden.
I want to show you how other terms come in,
which can couple two levels.
And also, actually when you drive transitions
within the hyperfine structure using radio frequency fields,
you're not driving them with the electric dipole approximation.
You're driving them with a magnetic dipole.
And this is actually the next approximation.
So there's a number of reasons why
I want to show you what the next order terms are
and how they actually lead to beautiful result,
magnetic dipole, and electric quadrupole transitions.

So our coupling term, just let me rewrite the equation.

Let me now simplify the notation by assuming
that the polarization is in the Z direction
and the propagation is in the X direction.
So then the coupling had the dipole term.
And the next order term is Ikx.
And this is what to investigate now.
The term, kx or kr is smaller.
And now I want to show you explicitly
that it's smaller by alpha.
The key vector of the photon one is related to the frequency hbar
omega divided by hbar c.
If I approximate for r, the relevant r
when we indicate over the wave function
will be the Bohr radius.
The relevant frequency, hbar omega, well, it's a Rydberg
and the Rydberg is e squared over the Bohr radius.
So now if I inserted, the Bohr radius cancels out.
And what we find is, well, kr is dimensionless.
We must find something which is dimensionless but expressed
by the fundamental constants of atomic physics.
And the only quantity which is available for that-- it's not
a surprise-- it's alpha, the fine structure constant.
So therefore, the dipole approximation
is actually the result of an expansion of the plain wave, e
to the i kr factor in units of for an extension
in the fine structure constant.

OK.
What we want to use-- we want to look now at the second term.

And so sometimes if you deal with a term,
we first make it more complicated
and then we simplify it.
I want to sort of summarize it and anti-summarize it
in the following way.
This is Pzx.
So let me substract z px.
But then of course I have to edit.

So now we have two terms.
One has a minus sign.
One has a plus sign.
And as we will see in just a minute,
is the first one is magnetic dipole transition.
The second one is electric quadrupole transition.

And we see that the first one can be regarded
as Pzx is like P dot r.
It's the y component or the vector product.
It is therefore the y component
of the orbital angular momentum.

OK.
So let's focus for now on this part,
the second term, which is the electric quadrupole
term we do in a few moments.
The relevant matrix element is now
the matrix element of the orbital angular momentum, ly.

The prefector-- let me just collect
all the constant imaginary unit-- e hbar AK over 2mc.

That looks complicated, but it immediately
simplifies when we realize that this here
is the Bohr magneton.
And while still have the vector potential--
but the magnetic field is the curl of the vector potential.
And that means for-- we assume that we have the vector
potential propagating in x, polarized
in z-- that means that our magnetic field is that.
So therefore, Ka, which appears in our expression
for the coupling, is just the magnetic field.
So therefore, we find the result that the next order
term of the coupling of that atoms
to the electromagnetic field have
the same form to the electromagnetic field
that it is the magnetic field part
of the electromagnetic wave, times the Bohr Magnetron
times the matrix element, due to the orbital angular momentum
operator.
And actually, if you take the orbital angular momentum
and multiply it with the Bohr magneton,
this is actually the operator of the magnetic moment rather
with a minus sign because the electron is negative charged.
Remember, we had introduced the operator
for the magnetic moment.
And the magnetic moment was the g factor
times the Bohr magneton, times the orbital angular momentum.
And the g factor for the orbital motion is one.
So therefore, the interaction we are talking about
is the Bohr Magnetron times the a magnetic field.

And of course, what we realize is the operator-- maybe
I should back up for a second and say, what we actually
realize is that the operator which
couples to the electromagnetic field
has actually formed U dot B. This is exactly what we
used for the Zeeman effect for the dc magnetic field.
But now the same form, U dot B appears
for time-dependent magnetic field.
And time-dependent magnetic fields
are not only creating level shifts,
they can also use transitions from the matrix element.
So in other words, this whole exercise
shows you that the form, U dot B, which appeared naturally
in the formulation for dc magnetic fields
also applies to ac magnetic fields.

But with that I can say, wait a moment.
There are now two sources for the magnetic moment
of the atom.
One is due to orbital angular momentum.
And the other one is due to spin angular momentum.
But the spin angular momentum has a g factor,
which is different, which the approximation
of the Dirac equation is 2.
So therefore, I mean, we will never
get spin out of a semi-classical discussion.
Remember, we started with a classical canonical treatment
of the electromagnetic field, canonical momentum.
And now we are running with it.
And we find that there's a coupling
term between the magnetic field and the magnetic moment
of the atom.
But of course, we only get the magnetic moment to the extent
that it comes from orbital motion.
But in a semi-classical way, I'm waving my hands now
and say well, what is ready for the coupling
to orbit to the magnetic moment of the angular momentum
also applies to the spin.
And I'm simply adding the spin here.
And with that, I've derived for you
the expression for the interaction matrix element
called m1.

m1 is magnetic dipole transitions.
I'll just write it down here.
So this is nothing else then U dot B. Any question?

We will just summarize-- we find it sort of interesting
when we discuss static electric and static magnetic fields.
For the static electric field, we
had an electric field times the dipole.
And we find this now as a time-dependent term
which can drive transitions for interactions
with time-dependent electromagnetic field.
But we find it in the dipole approximation.
The magnetic part, U dot B, we find
when we go to the next order.
We find it as a magnetic dipole term.
But there are more terms.
And I just want to illustrate it that and then I
stop with that multiple extension.
We had the second term, the kr term, or this one.
But then we sort of anti-summarize it.
The first one here, it could relate
to orbital angular momentum and to the magnetic moment.
And now I want to discuss the second term.
So that term uses a mixture of position and momentum
operators.
But we know already how we can get rid of momentum operators,
namely by expressing momentum operators as commutators
with the Hamiltonian.
So this gives us the commutator of z with h naught times x
plus z times the commutator of x with h naught.

OK.
So we have two commutators.
Each of them has two terms.
That means a total of four terms.
And if you write them down, you see that two are opposite
but equally and cancel.
And so therefore we are left with two terms,
which are minus H naught zx plus zx times H naught.

So this term has now the following contribution
to the coupling between the levels, a and b.

So we have this coupling matrix element.

We have prefactor mc km.

OK.
So using the same approach we used for the dipole matrix
element, the h naught can act on the state a on the right hand
and can act to understate b on the left hand side.
So this gives us simple the energy difference.
And what is left is now the matrix element,
which only involves position operators,
but it uses the product of 2.

And now again we express the vector potential
by the electric field as we had done before.
And therefore, we have now expressed our coupling,
yes, by an electric field.
And we assume that in the new regiment approximation
as we had used for the dipole coupling
that this is the frequency, omega.

OK.
So what we are realizing is that we
have one part of the interaction Hamiltonian, which
couples levels a and b, has-- we called it
e2, this electric quadrupole.
Because it involves elements of the quadrupole tensor,
a product of coordinates, xy, yz, xz, and for the geometry,
which we assume is our plane waves, it is zx.
It couples to the electric field.

And this is the prefactor.
For the more general geometry of plane waves
with different polarization going in different directions
we would have obtained different products or coordinates.
So let me indicate that what we have peaked out here
is one specific componenet of a tensor which
is sort of the tensor formed by using the position
vector, r, twice.
So in this derivation, we found that when
we take-- when we go beyond the dipole approximation,
when we take the kr term in leading order, that we
have two contributions.
One is m1, magnetic dipole.
The other one was electric quadrupole.
We realizee-- I tried to keep track of all the prefactors--
that the electric quadrupole matrix element is imaginary,
whereas the magnetic dipole was U dot B,
there was no imaginary unit.
It's real.
That means that you will never have any have any interference
effect between magnetic dipole and electric quadrupole.
In other words, when we have processes
like spontaneous emission, where we
take the square of the matrix element, the square
of the matrix element, it would be
the sum of the squares of the matrix element
for magnetic dipole and electric quadrupole transitions.